From: Jim Pryor <profjim@jimpryor.net>
Date: Sat, 16 Oct 2010 20:41:43 +0000 (-0400)
Subject: alternate Y1,Y2 tweak
X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=809b76fd4dfff7b2a54ebfd84a6f92ac6ae131ee;hp=8c28339b991f2d4f0940d295bf4d265a00751b05

alternate Y1,Y2 tweak

Signed-off-by: Jim Pryor <profjim@jimpryor.net>
---

diff --git a/hints/assignment_4_hint_3_alternate_1.mdwn b/hints/assignment_4_hint_3_alternate_1.mdwn
index 99a4202b..b0c06735 100644
--- a/hints/assignment_4_hint_3_alternate_1.mdwn
+++ b/hints/assignment_4_hint_3_alternate_1.mdwn
@@ -11,23 +11,43 @@ Alternate strategy for Y1, Y2
 
 	is implemented using regular, non-mutual recursion, like this (`u` is a variable not occurring free in `A`, `B`, or `C`):
 
-		let rec  u g x = (let f = u g in A)
-		in let rec g y = (let f = u g in B)
-		in let                f = u g in
+		let rec  u g x = (let f = u g in A)  in
+		let rec    g y = (let f = u g in B)  in
+		let                   f = u g        in
 		C
 
 	or, expanded into the form we've been working with:
 
-		let u = Y (\u g x. (\f. A) (u g)) in
-		let g = Y (  \g y. (\f. B) (u g)) in
-		let f =                     u g in
+		let u =    Y (\u g x. (\f. A) (u g))  in
+		let g =    Y (  \g y. (\f. B) (u g))  in
+		let f =                        u g    in
 		C
 
+	We abstract the Y1 and Y2 combinators from this as follows:
+
+		let Yu = \ff.    Y (\u g. ff ( u      g        ) g)  in
+		let Y2 = \ff gg. Y (  \g. gg (Yu ff   g        ) g)  in
+		let Y1 = \ff gg.             (Yu ff) (Y2 ff gg)      in
+		let f  = Y1 (\f g. A) (\f g. B)  in
+		let g  = Y2 (\f g. A) (\f g. B)  in
+		C
+
+
 *	Here's the same strategy extended to three mutually-recursive functions. `f`, `g` and `h`:
 
-		let u = Y (\u g h x.      (\f. A) (u g h)) in
-		let w = Y (  \w h x. (\g. (\f. B) (u g h)) (w h)) in
-		let h = Y (    \h x. (\g. (\f. C) (u g h)) (w h)) in
+		let v = Y (\v g h x.      (\f. A) (v g h)) in
+		let w = Y (  \w h x. (\g. (\f. B) (v g h)) (w h)) in
+		let h = Y (    \h x. (\g. (\f. C) (v g h)) (w h)) in
 		let g =                                     w h in
-		let f =                            u g h in
+		let f =                            v g h in
 		D
+
+	Or in Y1of3, Y2of3, Y3of3 form:
+
+		let Yv    = \ff.       Y (\v g h.      ff (    v g h) g h)               in
+		let Yw    = \ff gg.    Y (  \w h. (\g. gg (Yv ff g h) g h) (       w h))  in
+		let Y3of3 = \ff gg hh. Y (    \h. (\g. hh (Yv ff g h) g h) (Yw ff gg h))  in
+		let Y2of3 = \ff gg hh.                                      Yw ff gg (Y3of3 ff gg hh)  in
+		let Y1of3 = \ff gg hh.                     Yv ff (Y2of3 ff gg hh) (Y3of3 ff gg hh)  in
+		D
+